View of THE COUPLED-CLUSTER APPROACH TO QUANTUM MANY-BODY PROBLEM IN A THREE-HILBERT-SPACE REINTERPRETATION

doi:10.14311/AP.2014.54.0085 available online athttp://ojs.cvut.cz/ojs/index.php/ap

THE COUPLED-CLUSTER APPROACH TO QUANTUM MANY-BODY PROBLEM IN A THREE-HILBERT-SPACE

REINTERPRETATION

Raymond F. Bishop

, Miloslav Znojil

a School of Physics and Astronomy, Schuster Building, The University of Manchester, Manchester, M13 9PL, UK b Nuclear Physics Institute ASCR, 250 68 Řež, Czech Republic

∗ corresponding author: znojil@ujf.cas.cz

Abstract. The quantum many-body bound-state problem in its computationally successful coupled cluster method (CCM) representation is reconsidered. In conventional practice one factorizes the ground- state wave functions|Ψi=e^S|Φiwhich live in the “physical” Hilbert spaceH^(P) using an elementary ansatz for |Φi plus a formal expansion of S in an operator basis of multi-configurational creation operatorsC_⁺. In our paper a reinterpretation of the method is proposed. Using parallels between the CCM and the so called quasi-Hermitian,aliasthree-Hilbert-space (THS), quantum mechanics, the CCM transition from the known microscopic Hamiltonian (denoted by usual symbolH), which is self-adjoint inH^(P), to its effective lower-case isospectral avatar ˆh=e^−SHe^S, is assigned a THS interpretation. In the opposite direction, a THS-prescribed, non-CCM, innovative reinstallation of Hermiticity is shown to be possible for the CCM effective Hamiltonian ˆh, which only appears manifestly non-Hermitian in its own (“friendly”) Hilbert spaceH^(F). This goal is achieved via anad hocamendment of the inner product inH^(F), thereby yielding the third (“standard”) Hilbert space H^(S). Due to the resulting exact unitary equivalence between the first and third spaces,H^(P)∼ H^(S), the indistinguishability of predictions calculated in these alternative physical frameworks is guaranteed.

Keywords: quantum many-body problem, coupled cluster method,ad hocinner product, alternative representation spaces.

1. Introduction

The coupled cluster method (CCM) of construction, say, of the ground-state energies and wave functions of general quantum many-body systems works with vir- tual multi-particle excitations, and the linked-cluster nature of the contributions to the resulting estimates of measurable quantities is particularly emphasized [1]

– [3]. The strategy leads, in practical calculations, to the replacement of a given, known, realistic and exact microscopic input Hamiltonian (let us denote it by the dedicated symbolH) by its lower-case isospectral reparametrization

ˆh= Ω⁻¹HΩ. (1)

An optimal similarity-mediating transformation op- erator Ω is then sought in an exponential, mani- festly linked-cluster form Ω = expS. The excitations themselves are usually assumed multi-configurational, multi-indexed and generated by a complete set of mutually commuting many-body creation operators C_⁺ ≡(C_⁻)^† such that, conventionally, C₀⁺≡I and C₀⁻≡I whileS=P

6=0SC_⁺.

Naturally, the quality of the variationally deter- mined CCM coefficientsS translates into the quality of the predicted expectation values of any operator of an observable quantity. In practice, there expect- edly emerges a conflict between the precision and the costs of the results. One is thus forced to find an

optimal compromise between these two requirements by introducing various approximation schemes. In our present short paper we intend to describe one possible systematic approach to the abstract formulation of approximation hierarchies.

Our considerations will be inspired by the recent progress achieved in both the formal and the applied analyses of isospectral partnerships ˆh↔H. In par- ticular, we shall emphasize the innovative role played by various non-unitarymappings Ω, say, in their al- ternative time-independent or time-dependent forms as described in review papers [4] and [5], respectively.

Once a decisive simplification of the Hamiltonian is achieved by a non-unitary map Ω : H → ˆh, we have to start working with the less usual form ˆhof the Hamiltonian, which becomes, in general, non- Hermitian since

hˆ^†= Ω^†H(Ω⁻¹)^† = Ω^†ΩˆhΩ⁻¹(Ω⁻¹)^†,

Ω^†Ω≡Θ6=I. (2) In our present paper we intend to reveal and describe a deeper relationship between the CCM and the ab- stract framework provided by the mathematical the- ory of Hamiltonians exhibiting the above property of quasi-Hermiticity [6], aliascrypto-Hermiticity [7], with respect to the alternative Hilbert-space metric- operator Θ6=I,

ˆh^†Θ = Θˆh. (3)

In section 2 we shall explain the abstract formalism of three-Hilbert-space (THS) representation of quan- tum systems. We shall make use of the notation conventions of review paper [5], however, with the single, CCM-adapted exception of an interchange of the meaning of the lower- and upper-case symbols for the Hamiltonian. For the sake of clarity, Table 1 offers the explicit translation of the present notation conventions (as displayed in the first column) to the language of [5] (given in the second column). Sub- sequently, in section 3 an overall review of the key ideas of CCM constructions will be recalled, and their reinterpretation within the general THS scheme will be described. Section 4 will finally summarize our observations and proposals.

2. THS representation of a quantum system

2.1. Inspiration: Fourier transform The most elementary one-dimensional harmonic-oscil- lator Hamiltonian

H^(HO)=− d² dx² +x²

may be recalled as one of the best known examples of an operator representing a typical quantum observable.

It enters the ordinary differential Schrödinger equation H^(HO)ψ_n^(P)(x) =E_n^(HO)ψ^(P)_n (x),

ψ^(P)_n (x)∈L²(R), n= 0,1, . . . (4) for “physical” wave functionsψ^(P)n (x). The solution of this eigenvalue problem yields the well known discrete spectrum of bound-state energies E₀ = 1, E₁ = 3, E₂= 5, . . . , while the related wave functions belong to the most common Hilbert space of square-integrable complex functions ofx∈R. The argumentxof the wave functions coincides with an admissible value of the position of the quantum particle in question. In other words, the^(P)-superscripted complex functions ψn^(P)(x) may be interpreted as yielding the probability density of finding the particle at spatial pointx∈R. The wave functions in question live in a physical Hilbert spaceL²(R) ≡ H^(P). Formally, these func- tions may be represented as Fourier transforms of elements of a, supposedly, “friendlier” Hilbert space, ψn^(P) = Fψn^(F), ψ^(F)n ∈ H^(F). By construction, the latter space is alsoL²(R) but the physical meaning of the argumentp∈Rof the new wave functionsψn^(F)(p) is different. At the same time, the primary observable (i.e., the energy) remains unchanged.

In practice, the harmonic oscillator appears equally well represented inbothof the Hilbert spacesH^(P,F). Whenever one moves to a more complicated model, however, one may find that one of these spaces is preferable. In other words, a unitary-mapping- mediated transition to a potentially friendlier Hilbert

spaceH^(F)should be employed whenever it appears to lead, say, to a simplification of the calculation of the energies or of the wave functions.

We only have to add here that the same recommen- dation remains valid even for mappingsH^(P)↔ H^(F) which cease to be unitary. In this sense, our freedom of choosing between the upper- and lower-case Hamil- tonians as expressed in Eq. (1) may prove important, say, as a source of acceleration of the rate of conver- gence of various numerical or variational calculations (see, e.g., their review in [4]).

2.2. Non-unitary mappings Ω = expS Our present text is basically inspired by the recent growth in popularity of quantum models in which the ad hocnon-unitary isospectral transformations

H→ˆh= Ω⁻¹HΩ (5) perceivably simplify the Hamiltonian. Thus, Eq. (5) offers a path towards the feasibility of the evaluation of bound-state energies in complicated quantum sys- tems via an Ω-mediated transition from a complicated

“primary” Hilbert spaceH^(P)to a “friendlier” Hilbert spaceH^(F).

2.2.1. Crypto-Hermitian IBM method

One should distinguish between several non-equivalent applications of the above-outlined ideas. In one of the key references on the whole subject [4], the authors start from the knowledge of an overcomplicated H and from a qualified guess of a suitable simplification mapping Ω6= Ω^†−1

. For a persuasive illustration of the practical efficiency of such an approach the authors recalled the so-called interacting-boson-model (IBM) calculations of the spectra of heavy atomic nuclei.

Using the Dyson-Maleev choice of the boson-fermion mappings Ω^(Dyson) this strategy was found to lead to successful and particularly computation-friendly forms of variational predictions of the measured energy levels [8].

The key condition of applicability of the latter IBM recipe may be seen in the feasibility of construction of the ultimate “effective” Hamiltonian ˆhof Eq. (5). One arrives at a non-Hermitian operator in general, ˆh6= ˆh^†. It is worth adding that an exception may occur when the original self-adjoint HamiltonianH accidentally happens to commute with the operator-product sym- metry Π = Ω Ω^†; notice that Π6= Θ unless we restrict attention to the mere normal-operator mappings Ω such that Ω^†Ω = ΩΩ^†.

Whenever ˆh6= ˆh^†, the practical determination of the eigenvalues of the transformed Hamiltonian must remain easy and efficient. The reason is that in com- parison with standard methods, one must replace the usual single time-independent Schrödinger equation by the followingdoubletof conjugate eigenvalue problems

ˆh|Φni=E_n|Φni, hΦ˜_m|ˆh=E_mhΦ˜_m|,

n, m= 0,1, . . . (6)

Concept CCM [3] THS [5]

(realistic, microscopic) (Hermitian) (Hermitian inH^(P))

initial Hamiltonian H h

(non-unitary) (creation) (general invertible map)

transformation expS Ω :H^(F)→ H^(P)

(assumed simplified) (non-Hermitian)

non−Hermitian inH^(F) and Hermitian inH^(S)

transformed Hamiltonian ˆh=e^−SHe^S H= Ω⁻¹hΩ Table 1. Warning: opposite notation conventions.

using the respective action-to-the-right and action-to- the-left conventions.

Interested readers may consult review paper [5], in which a detailed discussion of further subtleties is given, first of all, for the far from trivial Heisenberg- representation-like cases in which the non-unitary mapping Ω is also permitted to vary with time.

2.2.2. PT-symmetric models

A reversal of the application of the simplification H →ˆhmay be found promoted in the overall context of relativistic quantum field theory. In this entirely different domain of physics, Bender and his coau- thors were the first who advocated an alternative philosophy of first choosing a sufficiently elementary non-Hermitian ˆh and of postponing the reconstruc- tion of the overcomplicated selfadjoint operator H, sometimes even indefinitely.

The initial move is due to Bender and Boettcher, who published, in 1998, an influential letter [9]. In this work they noticed that certain elementary non- Hermitian toy-model operators ˆhappeared to possess real and bound-state-like spectra, which were discrete, non-degenerate and bounded from below. In 2001, their observations were rigorously proved while, a few years later, some of these results were also comple- mented by approximate reconstructions of the neces- sary metric operator(s) Θ = Θ(ˆh) (cf., e.g., review [10] for details).

On a model-independent level these developments finally resulted in a fully consistent innovative THS strategy in which onestartsfrom a sufficiently elemen- tary lower-case (i.e., non-Hermitian) candidate for a

“realistic-model” Hamiltonian ˆh6= ˆh^†. Under a num- ber of assumptions (cf., e.g., reviews [4, 11–13]) one is then able to re-constructa suitable Hilbert-space mapping Ω = Ω(ˆh) and, via Eq. (1), also a self-adjoint, textbook-compatible isospectral avatarH =H^†of the Hamiltonian living in H^(P). In other words, from the initial knowledge of a quantum-dynamics-determining operator ˆhone is able to reconstruct, in principle at least, one or several tractable, textbook-compatible phenomenological quantum-mechanical and/or field- theoretical models.

Naturally, the initial choice of Hamiltonian ˆh6= ˆh^† acting in H^(F) should guarantee that the pair of

Schrödinger Eqs. (6) remains sufficiently easily solv- able. This requirement is not so easily satisfied. In practice people usually accept various independent and additional simplification assumptions, therefore.

Among them, a truly exceptional status belongs to the so calledPT-symmetry assumption or, more generally, to the assumption of the so called pseudo-Hermiticity property of ˆh(interested readers should consult, e.g., review [12] for more details).

2.2.3. Towards the complex energy spectra.

A third and still different implementation of the non- Hermitian-observable ideas is much older than the previous two. It may be traced back to the tradi- tional model-space projection technique of Feshbach in which one of the non-unitary mappings Ω and Ω⁻¹ is chosen as a projector so that the other one cannot exist. It is well known that the resulting simplified effective Hamiltonians are restricted to a subspace while becoming energy-dependent in general. In this sense, Feshbach’s effective Schrödinger Eqs. (6) arede factononlinear.

Such a case certainly lies outside the scope of our present considerations. Still, it is worth noting that there has recently emerged a number of papers in which the authors pointed out the existence of nu- merous links between the latter studies of resonances (i.e., of the quantum Hamiltonians possessing complex spectra) and their above-mentioned real-spectrum alternatives. Interested readers may consult, e.g., monograph [14] to see a number of newly discov- ered connections between the physics of Hermitian and/or non-Hermitian effective Hamiltonians and the related mathematics, which recommends, say, the use of the concepts of the Kato’s exceptional points, etc.

One should also point out that even in the recent physics-based and experiment-oriented studies of the real-spectrum pseudo-Hermitian andPT-symmetric models there has been a definite increase of interest in the interdisciplinary applications of the THS-related concepts of the spontaneousPT-symmetry breakdown and/or explanations of the exceptional-point-related phase-transition mechanisms connected with the loss of the reality of the spectrum (cf., e.g., the recent quantum-theory-related review paper [15], or a sample

[16] of a successful transfer of these ideas even beyond the realm of quantum theory itself).

3. THS interpretation of CCM constructions

Having passed through the extensive list of motivating considerations we are now getting very close to the key purpose of our present paper. For the construction of a concrete backward mapping Ω = Ω(H) in the CCM context we see that we might accept directly some of the THS constructive techniques. Naturally, in the CCM framework we encounter the possibility of extending its philosophy and its range beyond the ground-state constructions. For this purpose we may decide to experiment with various THS-inspired alter- natives to the basic (bi-)variational CCM ansätze.

In an introductory step let us return, therefore, to the IBM-motivated version of the THS approach, in which one assumes a full knowledge of the realistic, albeit prohibitively complicated, HamiltonianH = H^†, defined in some microscopic physical Hilbert space H^(P). A qualified guess or construction of Ω will be then vital for the success of computations, i.e., first of all, for the success of the practically tractable construction and solution of the pair of Schrödinger Eqs. (6).

3.1. Brief introduction to CCM constructions

In the CCM context, the generic, Dyson-inspired non- unitary mapping Ω^(CCM) has traditionally been con- sidered in the specific linked-cluster form of an expo- nential operator Ω^(CCM) = expS. In the literature (cf., e.g., [17] with further references) one may find a huge number of practical applications of the CCM strategy by which the ground-state wave functions are sought in the form of products

|Ψi=e^S|Φi. (7)

The ket vector|Φirepresents here a normalized state (usually called the model state or reference state), intended to be employed as a cyclic vector with re- spect to a complete set of mutually commuting multi- configurational creation operatorsC_⁺≡(C_⁻)^†. Our use of the special symbol  for the index indicates that this is a multi-index that labels the set of all many-particle configurations. In other words, states of the many-particle quantum system in question can be all written as superpositions of basis statesC_⁺|Φi.

Variational eigenkets (7) of the many-body self- adjoint HamiltonianH =H^† are conveniently written in terms of the specific CCMoperatoransatz

S =X

6=0

SC_⁺. (8)

The fundamental CCM replacement (7) of an unknown vector |Ψi by an unknown operator S is very well motivated from several independent points of view.

One of the motivations is inherited from Rayleigh- Schrödinger perturbation theory, in which, at a cer- tain stage of construction, theoperatorSchrödinger equation H|Ψi = E|Ψi in question is replaced by itssinglebra-vector projectionh0|H|Ψi=Eh0|Ψior, more generally, by afinitemultiplet of such projections h0j|H|Ψi=Eh0j|Ψi.

The key advantage of such a reduction lies in the possibility of a variationally optimal choice of the bra-vectors h0_j|. By contrast, the property of the Hermiticity of Hamiltonian H becomes, to a large degree, irrelevant. Thus, one transfers this experience to the CCM context by introducing a complementary, formally redundant concept of left-action variational eigenvectorhΨ|˜ ofH.

The nontrivial difference between the tilded and untilded eigenvector|Ψi˜ and|Ψiis motivated by the possibility of introducing an additional set {S˜} of free parameters in the bra-vector

hΨ|˜ =hΦ|Se˜ ^−S; S˜=I+X

6=0

S˜C_⁻. (9)

Together with the conditions of completeness of the basis

X



C_⁺|ΦihΦ|C_⁻=I=|ΦihΦ|+X

6=0

C_⁺|ΦihΦ|C_⁻,

(10) and together with the usual properties of the creation and annihilation operators,

C_⁻|Φi= 0 =hΦ|C_⁺; ∀6= 0 (11) and

[C_⁺, C_J⁺] = 0 = [C_⁻, C_J⁻] (12) we arrive at the standard version of the CCM formal- ism, in which one currently employs approximations which do not make use of the manifest Hermiticity of the original eigenvalue problem. Such approxima- tions may entail keeping only a physically motivated subset of the multi-indices in the otherwise exact expansions of the correlation operatorsS and ˜S in Eqs. (7)–(9).

As an immediate mathematical consequence, the CCM Schrödinger equation for ground state acquires the two different and mutually non-conjugate alterna- tive forms

ˆh|Φi=E|Φi, hΦ|S˜ˆh=EhΦ|S,˜

ˆh= e^−SHe^S. (13) Obviously, once the two sets of coefficients{S}and {S˜}are determined, all the ground-state properties of the many-body system in question may be considered as known.

The ground-state expectation value of any given operator Λ should be evaluated from the asymmetric prescription

hΨ|Λ˜ |Ψi=hΦ|Se˜ ^−SΛe^S|Φi= ¯Λ(S_,S˜_). (14)

This recipe keeps trace of the artificial asymmetry as introduced in Eq. (13) which, in its turn, simplifies certain technical aspects of the global CCM approach.

In particular, in the bi-variational spirit the energy expectation formula

hΨ|H˜ |Ψi=hΦ|˜ ˆh|Φi (15) may now be minimized with respect to the full set of parameters {S,S˜}. Two equations follow, viz.,

hΦ|C_⁻ˆh|Φi= 0 ; ∀6= 0 (16) and

hΦ|S(ˆ˜ h−E)C_⁺|Φi= 0 ; ∀6= 0. (17) In their turn, these relations may be interpreted as a coupled algebraic set of equations that determine the parameters {S,S˜}. The consistency of the recipe may be reconfirmed by the derivation of the former relation (16) from the assumption of completeness of the set of states{hΦ|C_⁻}. Similarly, Eq. (17) may be perceived as a consequence of the completeness of the conjugate set{C_⁺|Φi}.

The coupled equations (16) and (17) are of the Goldstone linked-cluster type. For this reason, all extensive variables, such as the energy, scale linearly with the number of particles at every level of approxi- mation. This is another merit of the CCM construc- tion. Among the disadvantages we mention that the ground-state energy formula does not necessarily pro- vide an upper bound, due to the intentional violation of manifest Hermiticity for the problem. Still, the recipe enables us to determine both the quickly con- vergent energies as well as the Hamiltonian-dependent values of parameters{S,S˜} or, in various approxi- mate schemes, of the respective truncated subsets of these values.

Within the general framework of the CCM treat- ment of many-body quantum systems some of the above-mentioned assumptions and restrictions may be removed. The method may certainly be extended, say, to cover also excited states and/or certain time- dependent versions of dynamics. In both of these directions, an implementation of ideas from THS con- text might prove particularly helpful.

3.2. CCM–THS correspondence

The close mathematical relationship between the vari- ous variational CCM recipes and the universal three- Hilbert-space (THS) representation of a generic quan- tum system has been largely overlooked till now.

Apart from a few rather inessential differences, one of the key obstacles may be seen in the differences in their notations, a first sample of which is displayed in Table 1, where we see that for Hamiltonians, the CCM and THS notation conventions are strictly opposite (so we have to re-emphasize that in our present paper we are using the first-column notation conventions).

With due care paid to the Hermiticity or non- Hermiticity of the Hamiltonian, it seems equally im- portant to spot the CCM - THS coincidences and/or differences in the definitions and meanings of the other concepts. For the ground-state wave functions, in par- ticular, the parallels in the denotation of the same feature or quantity are displayed in Table 2.

An inspection of Table 2 reveals that in their respec- tive current versions, the two formalisms are far from equivalent, indeed. At the same time, they may be both found to suffer of certain specific weak points. In fact, our present considerations were originally moti- vated precisely by aparallelanalysis of these respective weaknesses. After their deeper study we came to the conclusion (documented and emphasized also by the above two respective compact reviews) that a perceiv- able profit might be gained by modifying and getting those two formalisms and/or methods of calculation closer to each other.

On the side of the CCM formalism, for exam- ple, one may immediately notice an obvious con- trast between the exponential CCM form of the map- ping Ω^(CCM)= expS^(CCM) and the manifestlynon- exponential, polynomial form of the tilded operator S˜ entering the second CCM ansatz (9). Naturally, such a striking difference did not stay unnoticed in the related literature, and the idea has been imple- mented into the so called extended version of the CCM (ECCM) formalism [1, 2].

On the side of the general THS formalism, in par- allel, we may now recollect one of the very popular formalism-simplifying tricks by which one works just with the special Hermitian mappings Ωs = Ω^†_s = expS_s [11, 12]. Under this additional assumption one arrives at a fairly natural exponential form of the equally special but still sufficiently general sub- set of the positive-definite metrics, Θs = exp 2Ss. In this manner, after the respective replacements S˜ →S˜^(ECCM) and Θ→Θs = exp 2Ss, the initially very different forms of the operators get closer.

Once one stops feeling discouraged by the similar, more or less purely formal differences, one has to re- open also the question of the respective roles of the operators ˜S and Θ in the purely numerical context.

This is another type of difference which is, naturally, strongly dependent on the purpose of the calculation.

Traditionally, the CCM and THS calculation purposes are truly rather different. Nevertheless, on the CCM side one immediately notices that the predominance of calculations of the ground-state characterstics does not exclude extensions, say, to the excited-state prob- lem [18] or even to the description of systems which are allowed to exhibit a manifest time-dependence of their dynamics [19]. In this sense we are getting still closer to the respective time-independent and time-dependent non-Hermitian versions of the general and universal THS formulation of abstract quantum mechanics as summarized, say, in Refs. [4] and [5].

Ground state CCM [3] THS [5]

Purpose bi-variationality re-Hermitization ofH in H^(S) Assumptions S˜ = annihilation Θ = Ω^†Ω, Ω = invertible eigen-ket (simplified) |Φi |0i ∈ H^(F,S)

eigen-bra (conjugate) hΦ| h0| ∈ H^(F)0

eigen-bra (amended) hΦ|˜ :=hΦ|S˜ hh0|:=h0|Θ ∈ H^(S)0 microscopic ground state |Ψi |0 = Ω|0i ∈ H^(P) first variational ansatz =e^S|Φi

left ground state hΨ|˜ ≺0| =h0|Ω^†=hh0|Ω⁻¹∈ H^(P)0 second variational ansatz =hΦ|e˜ ^−S

Table 2. Parallel notation conventions.

initial, givenmicroscopic HamiltonianH =H^† lives in primary spaceH^(P); all isprohibitively complicated:

•one constructsthe CCM operator Ω = expS

CCM mapΩ⁻¹= exp (−S) . &-equivalence

friendly spaceH^(F)isfalse:

• • in it,newˆh:= Ω⁻¹HΩ is not self-adjoint, ˆh6= ˆh^†

hermitization

−→

secondary spaceH^(S)isstandard:

• • •in it, the samehˆ= Θ⁻¹ˆh^†Θ := ˆh^‡ is found self-adjoint and diagonalizable Figure 1. The three-Hilbert-space diagram.

4. Discussion

4.1. A CCM–THS fusion?

In the language of mathematics the core of our present message may be summarized as follows: in fact, it need not be particularly difficult to search for a further enhancement of parallels between the manifestly non- Hermitian, annihilation-operator-type CCM choice of the tilded operator ˜S and the strictly Hermitian and, in addition, also strictly positive definite Hilbert- space-metric operator Θ = Ω^†Ω. In the terminology of physics this persuasion is supported by the obser- vation that what issharedby both theabstractCCM and THS formalisms is a truly exciting idea of using nontrivial“redundant” operators ˜S or Θ in place of the common identity operator.

In both formalisms, the rationale behind the use of the respective nontrivial operators ˜S and Θ is rather subtle though fairly persuasive and not too dissimilar.

Indeed, one starts from a well known while, unfortu- nately, prohibitively complicated initial self-adjoint Hamiltonian in both cases (recall, once more, Table 1). Secondly, the choice and/or construction of the mapping Ω = expS is motivated, in both of the ap- proaches, by a more or less comparably successful simplificationof the Schrödinger eigenvalue problem.

Thirdly, both the CCM and THS re-arrangements of the quantum bound-state problem lead to the neces-

sity of the introduction of the respective nontrivial operators ˜S and Θ usingcomparably strong but, at the same time,differentsupportive arguments.

What now remains open is a truly challenging ques- tion as to whether, and in which sense, one could really achieve a complete coincidence of the respec- tive (and, apparently, ideologically distant) CCM and THS recipes. Firstly, an affirmative answer may be given (and the idea may be made working) when- ever the Hilbert spaces of the system remain, for whatever reason (e.g., for approximation purposes) finite-dimensional.

In such a very specific case the space for a com- promise immediately opens after we move from the abstract formalism to any kind of a practical varia- tional calculation and/or numerical approximation.

Schematically speaking, any 2M-parametric array of the multi-indexed CCM variational coefficients S_k

and ˜S_k withk= 1,2, . . . , M may be perceived equiv- alent to an introduction of a 2M-parametric metric Θ = Ω^†Ω. It should be noted, as a supportive ar- gument, that even in the thorough IBM review [4]

a large amount of space has been devoted to the study of finite-dimensional models and to the questions of practical variational applicability of the THS scheme.

On this level of mathematics the overall nature and structure of the above-indicated possibility of a com- plete unification (or, at least, of a strengthening of

the CCM–THS parallelism) may be read out of the three-Hilbert-space diagram in Figure 1. By the blobs we mark here the three main constructive CCM–THS steps. In the first two steps (viz.,• and••) we may assume to stay inside the usual CCM framework in which the ground-state eigenvector|Ψiof the quan- tum system in question is reparametrized in terms of operatorS. Thus, the CCM–THS innovation only emerges, via operators ˜S alias Θ, in the third step (• • •, see Table 2).

In this setting let us remind the readers that the (certainly, in general, existing) creation-operator com- ponents of Θ^(CCM) may be expected to play just a marginal role in the convergence. The reason is that the CCM choice of Ω = expS is mainly aimed at the construction of the many-body ground states. Thus, a lot of freedom is left for the introduction of more variational parameters via ˜S 6=I. In contrast, the balanced distribution of attention of the universal THS formulae between the ground and excited states lowers, certainly, the latter freedom because the THS recipe defines the metric in terms of Ω unambiguously.

4.2. Towards the infinite-dimensional Hilbert spaces

Once we decide to leave the language of computing and once we move to the exact description of realistic quantum systems and to the (say, separable) infinite- dimensional Hilbert spaces, the search for the CCM–

THS unification becomes perceivably more difficult.

From the THS perspective, in particular, the key subtlety lies in the fact that whenever one decides to treat the two topological vector spacesH^(P)andH^(F) (naturally, still without any account of the definition of the inner products and of the metrics) as distinct, the map Ω = expS will slightly change its meaning as well as its interpretation.

From the alternative (and also historically older) CCM point of view it is necessary to recall, first of all, the results of the important paper [20]. Its author ac- cepted the usual, above-described CCM linked-cluster parametrization, in its most general time-dependent form, as deduced from an appropriate action principle.

In turn, this enforces a symplectic structure on the ensuing CCM phase space of the real-valued “degrees of freedom”S,S˜ of Eq. (14).

At this point the author of [20] has been forced to discuss the emergence of the characteristic non- Hermiticity of the average-value functionals ¯Λ(S,S˜) of physical observables as well as of the action A(S¯ ,S˜) itself.

In fact, our present idea of possible CCM–THS cor- respondence also found another source of inspiration in his approach, so let us recall his key ideas in more detail. Firstly, he introduced the set of complex conju- gate variablesS_^∗,S˜_^∗ and showed how they could be used to enlarge the CCM phase space into a genuine complex manifold but of too large a dimensionality.

He further showed how the extra degrees of freedom

could then be eliminated via the Dirac bracket tech- nique. A set of constraint functions was introduced which thereby select the physical submanifold (alias the reduced phase space, or constraint surface) corre- sponding to the original Hilbert space. Subsequently, the reduced phase space was shown to be a (Kähler) complex manifold with a symplectic structure, just as the original extended one.

Ultimately, the Kähler manifold may be perceived as defining a positive, invertible, Hermitian geometry in the reduced phase space. Arponen [20] further shows that for a compound operator productQ= Λ1Λ2, the CCM star product which generates the expectation- value functionalQ= Λ1Λ2 in terms of the individual expectation values Λ1 and Λ2, as given by Eq. (14), can be well defined in the reduced (i.e., physical) phase space.

This result suggests that besides starting from the THS scheme, one could also try to develop certain innovative and consequently Hermiticity-preserving hierarchical approximation schemes strictly within the CCM framework. A judicious use of the on-shell star products seems capable of establishing another form of the CCM–THS parallels, and of doing so in an entirely general setting. In addition, some explicit and concreteconstructiveimplementations of the concept of the metric Θ may be found directlyin the generic CCM framework. Naturally, a deeper analysis would require a verification in terms of explicit constructions.

Further development of such a project lies, naturally, beyond the scope of our present paper.

4.3. Outlook

Let us summarize that in the general THS framework one is expected to perform all of the practical compu- tations of physical predictions inside the “friendliest”

Hilbert space H^(F). What is a real mathematical promise of a search for the new mutual CCM–THS correspondences is that even the standard probabilis- tic interpretation of many-body wave functions need not require a return to the “unfriendly” space H^(P). In all respects it becomes easier to replace the latter space by its (unitarily) equivalent alternative H^(S). The reason is that the latter Hilbert space only differs from its more friendly predecessor H^(F) by anad hoc amended inner product.

Our present brief outline of a few explicit CCM–

THS correspondences centered around the fact that operator ˜S of the CCM formalism coincides with the Hilbert-space metric operator Θ after a “translation of notation” to the THS-representation language of [5]. On the background of this comparison the main potential innovation of the CCM was found in the THS-based possibility of distinguishing between the three separate Hilbert spaces H^(P), H^(F) andH^(S), which would representthe samequantum many-body system.

The change of perspective revealed several CCM–

THS parallels as well as differences. Among the par-

allels, one of the most inspiring seems to lie in the emerging structural similarity between the CCM con- structions and their IBM (= interacting boson model) counterparts. The project of our future development of such a CCM–IBM correspondence seems promis- ing. In the language of physics it might enable us to keep the initial physicalP-superscripted Hilbert space as fermionic while rendering the other two,F- and S-superscripted Hilbert spaces, strictly in the general- ized IBM spirit, carriers of another, generalized (e.g., pseudo-bosonic) statistics.

In the opposite direction, also the traditional IBM constructions of effective Hamiltonians could find some new inspiration in their CCM analogues. In particular, the prospects of a simplification mediated by the non- unitary invertible mappings Ω = expS need not nec- essarily stay bound by their traditional bosonic-image IBM restrictions. A new wealth of correspondences may be expected to become implementable between the auxiliary Hilbert spaceH^(F)and the, by assump- tion, prohibitively complicated physical Hilbert space H^(P) (hence, the superscript^(P)may also mean “pro- hibitive”).

Ultimately, the technically most productive idea may be seen in the exceptional role of the F- superscripted Hilbert space, in which the absence of an immediate physical interpretation (say, of the measurable aspects of coupled clusters) appears more than compensated by the optimal suitability of this particular representation space for calculations of the, typically, variational CCM type.

Acknowledgements

The participation of MZ was supported by GAČR grant P203/11/1433.

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